Calculate Next tile needed given a zoom level

I am working on an app that generates a map based on certain map features. Given LatLon coordinates and a zoom level, how do I determine how far to offset my next point in a given direction to grab tile next to the tile I just grabbed? I'm trying to
minimize the number of tiles that I need to grab.

Also, is there a way to calculate the changes in lat/long for each pixel given the zoom level?